Topology and its Applications最新文献

{"title":"Various notions of shadowing in triangular system and its component systems","authors":"Deepanshu Dhawan,&nbsp;Puneet Sharma","doi":"10.1016/j.topol.2024.109109","DOIUrl":"10.1016/j.topol.2024.109109","url":null,"abstract":"<div><div>In this article, we investigate various forms of shadowing for a general triangular system. In particular, we relate various notions of shadowing for a triangular system with various notions of shadowing in the component systems. We prove that if the base <em>f</em> for <em>T</em> is transitive then shadowing in the base map and the non-autonomous system generated by a transitive point ensures shadowing of the triangular system. We prove that if the base map for <em>T</em> is expansive then shadowing in the triangular system ensures shadowing in the component systems. We prove that if the non-autonomous component systems form a synchronized family and the base map possesses a globally attracting fixed point <span><math><msub><mrow><mi>x</mi></mrow><mrow><mn>0</mn></mrow></msub></math></span> then eventual shadowing in system generated by <span><math><msub><mrow><mi>x</mi></mrow><mrow><mn>0</mn></mrow></msub></math></span> ensures eventual shadowing in each of the non-autonomous component systems. We also investigate chain transitivity and chain mixing for a general triangular system.</div></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"357 ","pages":"Article 109109"},"PeriodicalIF":0.6,"publicationDate":"2024-10-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142534568","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Non-abelian tensor product and circular orderability of groups","authors":"Maxim Ivanov","doi":"10.1016/j.topol.2024.109111","DOIUrl":"10.1016/j.topol.2024.109111","url":null,"abstract":"<div><div>For a group <em>G</em> we consider its tensor square <span><math><mi>G</mi><mo>⊗</mo><mi>G</mi></math></span> and exterior square <span><math><mi>G</mi><mo>∧</mo><mi>G</mi></math></span>. We prove that for a circularly orderable group <em>G</em>, under some assumptions on <span><math><msub><mrow><mi>H</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>(</mo><mi>G</mi><mo>)</mo></math></span> and <span><math><msub><mrow><mi>H</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>(</mo><mi>G</mi><mo>)</mo></math></span>, its exterior square and tensor square are left-orderable. This yields an obstruction for a circularly orderable group <em>G</em> to have torsion. We apply these results to study circular orderability of tabulated virtual knot groups.</div></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"358 ","pages":"Article 109111"},"PeriodicalIF":0.6,"publicationDate":"2024-10-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142552383","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"An upper bound for the number of critical points of the systole function on the moduli space of hyperbolic surfaces","authors":"Yue Gao","doi":"10.1016/j.topol.2024.109091","DOIUrl":"10.1016/j.topol.2024.109091","url":null,"abstract":"<div><div>We obtain an upper bound for the number of critical points of the systole function on <span><math><msub><mrow><mi>M</mi></mrow><mrow><mi>g</mi></mrow></msub></math></span>. Besides, we obtain an upper bound for the number of those critical points whose systole is smaller than a constant.</div></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"358 ","pages":"Article 109091"},"PeriodicalIF":0.6,"publicationDate":"2024-10-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142530753","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"On sequences of finitely supported measures related to the Josefson–Nissenzweig theorem","authors":"Witold Marciszewski ,&nbsp;Damian Sobota ,&nbsp;Lyubomyr Zdomskyy","doi":"10.1016/j.topol.2024.109100","DOIUrl":"10.1016/j.topol.2024.109100","url":null,"abstract":"<div><div>Given a Tychonoff space <em>X</em>, we call a sequence <span><math><mo>〈</mo><msub><mrow><mi>μ</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>:</mo><mi>n</mi><mo>∈</mo><mi>ω</mi><mo>〉</mo></math></span> of signed Borel measures on <em>X a finitely supported Josefson–Nissenzweig sequence</em> (in short <em>a JN-sequence</em>) if: 1) for every <span><math><mi>n</mi><mo>∈</mo><mi>ω</mi></math></span> the measure <span><math><msub><mrow><mi>μ</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> is a finite combination of one-point measures and <span><math><mo>‖</mo><msub><mrow><mi>μ</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>‖</mo><mo>=</mo><mn>1</mn></math></span>, and 2) <span><math><msub><mrow><mo>∫</mo></mrow><mrow><mi>X</mi></mrow></msub><mi>f</mi><mi>d</mi><mspace></mspace><msub><mrow><mi>μ</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>→</mo><mn>0</mn></math></span> for every continuous function <span><math><mi>f</mi><mo>∈</mo><mi>C</mi><mo>(</mo><mi>X</mi><mo>)</mo></math></span>. Our main result asserts that if a Tychonoff space <em>X</em> admits a JN-sequence, then there exists a JN-sequence <span><math><mo>〈</mo><msub><mrow><mi>μ</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>:</mo><mi>n</mi><mo>∈</mo><mi>ω</mi><mo>〉</mo></math></span> such that: i) <span><math><mi>supp</mi><mo>(</mo><msub><mrow><mi>μ</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>)</mo><mo>∩</mo><mi>supp</mi><mo>(</mo><msub><mrow><mi>μ</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>)</mo><mo>=</mo><mo>∅</mo></math></span> for every <span><math><mi>n</mi><mo>≠</mo><mi>k</mi><mo>∈</mo><mi>ω</mi></math></span>, and ii) the union <span><math><msub><mrow><mo>⋃</mo></mrow><mrow><mi>n</mi><mo>∈</mo><mi>ω</mi></mrow></msub><mi>supp</mi><mo>(</mo><msub><mrow><mi>μ</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>)</mo></math></span> is a discrete subset of <em>X</em>. We also prove that if a Tychonoff space <em>X</em> carries a JN-sequence, then either there is a JN-sequence <span><math><mo>〈</mo><msub><mrow><mi>μ</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>:</mo><mi>n</mi><mo>∈</mo><mi>ω</mi><mo>〉</mo></math></span> on <em>X</em> such that <span><math><mo>|</mo><mi>supp</mi><mo>(</mo><msub><mrow><mi>μ</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>)</mo><mo>|</mo><mo>=</mo><mn>2</mn></math></span> for every <span><math><mi>n</mi><mo>∈</mo><mi>ω</mi></math></span>, or for every JN-sequence <span><math><mo>〈</mo><msub><mrow><mi>μ</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>:</mo><mi>n</mi><mo>∈</mo><mi>ω</mi><mo>〉</mo></math></span> on <em>X</em> we have <span><math><msub><mrow><mi>lim</mi></mrow><mrow><mi>n</mi><mo>→</mo><mo>∞</mo></mrow></msub><mo>⁡</mo><mo>|</mo><mi>supp</mi><mo>(</mo><msub><mrow><mi>μ</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>)</mo><mo>|</mo><mo>=</mo><mo>∞</mo></math></span>.</div></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"364 ","pages":"Article 109100"},"PeriodicalIF":0.6,"publicationDate":"2024-10-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143529548","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Rigid topologies on groups","authors":"Eli Glasner ,&nbsp;Benjamin Weiss","doi":"10.1016/j.topol.2024.109094","DOIUrl":"10.1016/j.topol.2024.109094","url":null,"abstract":"<div><div>Our main result is to show that every infinite, countable, residually finite group <em>G</em> admits a Hausdorff group topology which is neither discrete nor precompact.</div></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"364 ","pages":"Article 109094"},"PeriodicalIF":0.6,"publicationDate":"2024-10-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143529101","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Banakh spaces and their geometry","authors":"Taras Banakh","doi":"10.1016/j.topol.2024.109105","DOIUrl":"10.1016/j.topol.2024.109105","url":null,"abstract":"<div><div>Following Will Brian, we define a metric space <em>X</em> to be <em>Banakh</em> if all nonempty spheres of positive radius <em>r</em> in <em>X</em> have cardinality 2 and diameter 2<em>r</em>. Standard examples of Banakh spaces are subgroups of the real line. In this paper we study the geometry of Banakh spaces, characterize Banakh spaces which are isometric to subgroups of the real line, and also construct Banakh spaces <em>X</em> which do not embed into the real line and have a prescribed distance set <span><math><mi>d</mi><mo>[</mo><msup><mrow><mi>X</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>]</mo></math></span>.</div></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"364 ","pages":"Article 109105"},"PeriodicalIF":0.6,"publicationDate":"2024-10-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143529551","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Diagonals of the square of a continuum","authors":"A. Illanes ,&nbsp;V. Martínez-de-la-Vega ,&nbsp;Jorge M. Martínez-Montejano ,&nbsp;Daria Michalik","doi":"10.1016/j.topol.2024.109099","DOIUrl":"10.1016/j.topol.2024.109099","url":null,"abstract":"<div><div>Given a continuum <em>X</em>, let <span><math><msup><mrow><mi>X</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>=</mo><mi>X</mi><mo>×</mo><mi>X</mi></math></span> and <span><math><mi>Δ</mi><mo>=</mo><mo>{</mo><mo>(</mo><mi>x</mi><mo>,</mo><mi>x</mi><mo>)</mo><mo>∈</mo><msup><mrow><mi>X</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>:</mo><mi>x</mi><mo>∈</mo><mi>X</mi><mo>}</mo></math></span>. We deal with the following properties of Δ in <span><math><msup><mrow><mi>X</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span>: the colocal connectedness, non-weak cut, non-block, and non-cut. We consider the following classes of continua: arcwise connected, locally connected, dendroids, indecomposable continua and metric compactifications of the ray <span><math><mo>[</mo><mn>0</mn><mo>,</mo><mo>∞</mo><mo>)</mo></math></span>.</div></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"364 ","pages":"Article 109099"},"PeriodicalIF":0.6,"publicationDate":"2024-10-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143529545","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Group topologies on the integers generated by sequences","authors":"Lydia Außenhofer ,&nbsp;Dikran Dikranjan","doi":"10.1016/j.topol.2024.109102","DOIUrl":"10.1016/j.topol.2024.109102","url":null,"abstract":"<div><div>In the first part of the paper we study various ways to introduce a group topology on the group <span><math><mi>Z</mi></math></span> of integers by using sequences of natural numbers <strong>b</strong>, among them the well-known Protasov-Zelenyuk topology <span><math><msub><mrow><mi>T</mi></mrow><mrow><mi>b</mi></mrow></msub></math></span> for a <em>T</em>-sequence <strong>b</strong>. We prove, among others, that for slowly diverging sequences <strong>b</strong>, the group <span><math><mo>(</mo><mi>Z</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>b</mi></mrow></msub><mo>)</mo></math></span> is not locally quasi-convex, in particular not reflexive. As an application, in the second part we resolve Problem 6.1 and Problem 6.5 from <span><span>[28]</span></span>.</div></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"364 ","pages":"Article 109102"},"PeriodicalIF":0.6,"publicationDate":"2024-10-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143529546","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Accept/Reject games","authors":"Nadav Samet ,&nbsp;Boaz Tsaban","doi":"10.1016/j.topol.2024.109108","DOIUrl":"10.1016/j.topol.2024.109108","url":null,"abstract":"<div><div>Many properties of topological spaces, like compactness, have the form that any cover of the space of a certain type contains a subcover of some other type. We introduce a game theoretic analogue of this selection prototype, the <em>Accept/Reject game</em>, and relate it to classical games and selection principles. Modified versions of this game are shown to be equivalent to standard selection principles games.</div></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"364 ","pages":"Article 109108"},"PeriodicalIF":0.6,"publicationDate":"2024-10-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143529222","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Continuous Lebesgue measure-preserving maps on one-dimensional manifolds: A survey","authors":"Jozef Bobok ,&nbsp;Jernej Činč ,&nbsp;Piotr Oprocha ,&nbsp;Serge Troubetzkoy","doi":"10.1016/j.topol.2024.109101","DOIUrl":"10.1016/j.topol.2024.109101","url":null,"abstract":"<div><div>We survey the current state-of-the-art about the dynamical behavior of continuous Lebesgue measure-preserving maps on one-dimensional manifolds.</div></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"364 ","pages":"Article 109101"},"PeriodicalIF":0.6,"publicationDate":"2024-10-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143529549","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}